Problem 56 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 7(2 x+5)-4(x+2)-20 x $$

Step-by-Step Solution

Answer

The simplified expression is \(-10x + 27\).

Step 1: Distribute the coefficients inside the parentheses

Start by distributing the coefficients across each term within the parentheses. For the expression \(7(2x + 5)\), distribute 7, resulting in \(14x + 35\). For \(-4(x + 2)\), distribute -4, resulting in \(-4x - 8\). So, the expression becomes: \[ 14x + 35 - 4x - 8 - 20x \]

Step 2: Combine like terms

Now, combine the like terms in the expression. Identify the terms with \(x\): these are \(14x\), \(-4x\), and \(-20x\). Add these terms together:\[ 14x + (-4x) + (-20x) = -10x \] Next, combine the constant terms \(35\) and \(-8\):\[ 35 - 8 = 27 \]The expression now simplifies to:\[ -10x + 27 \]

Key Concepts

Distributive Property

The distributive property is an important tool in algebra that helps in breaking down expressions into simpler parts. It is the key to removing parentheses in mathematical expressions. When you have an expression like \( a(b + c) \), the distributive property lets you multiply \( a \) by each term within the parentheses.

Here's how it works:

• Multiply the outside number or term with each term inside the parentheses separately.
• Add or subtract these results to get a new expression without parentheses.

For example, in the expression \( 7(2x + 5) \), you apply the distributive property:

• Multiply \( 7 \) by \( 2x \), resulting in \( 14x \).
• Then, multiply \( 7 \) by \( 5 \), resulting in \( 35 \).

You get \( 14x + 35 \). This step prepares the expression for further simplification.

Combining Like Terms

Combining like terms is the next step after using the distributive property. This process helps simplify expressions by merging terms that have the same variables. Like terms are terms that contain the same variable raised to the same power.

For example, in the expression \( 14x + (-4x) + (-20x) \), each term is a 'like term' because they all contain the variable \( x \). To combine them, you:

• Add and subtract their coefficients: \( 14 + (-4) + (-20) \).
• This equals \( -10 \), giving us the term \( -10x \).

Next, combine the constant terms, like \( 35 \) and \( -8 \):

• Their sum is \( 27 \).

By combining like terms, you simplify the expression to \( -10x + 27 \). This step is crucial for achieving the simplest form of an algebraic expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). They form the building blocks of algebraic equations and need to be simplified to better understand or solve them.

An expression like \( 7(2x + 5) - 4(x + 2) - 20x \) involves several components:

• Constants, which are standalone numbers like \( 7 \), \(-4 \), and \(-20 \).
• Variables, represented with letters like \( x \) in this case.
• Operations, which are addition, subtraction, and multiplication represented by symbols like \(+ \), \(- \), and implicitly in \( 7(2x + 5) \).

Simplifying such expressions allows for easier computation and helps in solving equations derived from real-world problems. It involves using algebraic techniques such as the distributive property and combining like terms to achieve a form that is straightforward to interpret or solve. Understanding these components and their simplification is key in mastering algebra.